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examThreeReview

9 determine if the sequence a n converges if it

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9. Determine if the sequence { a n } converges. If it converges, find its limit. (a) a n = 1 + ( - 1) n n (b) a n = 100 n 2 + 2 billion gajillion 2 n 3 - 7 (c) a n = e 1 /n (d) a n = ln(2 /n 2 ) (e) a n = n ! 2 n (f) a n = k n n ! for some real number k . (g) a n = tan(2 /n ) (h) a n = ln( n 2 e n ) (i) a n = 5 + n n + ( - 1) n sin( ) (j) a n = cos parenleftBig πn 2 parenrightBig (k) a n = n 2 e n (l) a n = parenleftbigg 1 - 2 n parenrightbigg n 10. Find the general term of the sequence. (a) 1 2 , 3 4 , 5 6 , 7 8 , . . . (b) 0 , 1 4 , 2 9 , 3 16 , . . . (c) 1 - 1 2 , 1 2 - 1 3 , 1 3 - 1 4 , . . . (d) 1 3 , 2 4 , 3 5 , 4 6 , . . . (e) 1 3 , 4 5 , 9 7 , 16 9 , . . . (f) 2 , 2 , 2 , . . . (g) 0 , 2 , 0 , 2 , 0 , 2 , . . . (h) - 1 , 1 4 , - 1 9 , 1 16 , . . . (i) 2 4 , 4 7 , 6 12 , 8 19 , 10 28 , . . . 11. (a) Show that 1 + 2 + 3 + · · · + ( n - 1) + n = n ( n + 1) 2 (b) Find the sum of all numbers between (and including) 1 and 100. (c) Find the limit of the sequence a n = 1 n 2 + 2 n 2 + 3 n 2 + · · · + n n 2 12. Recall that the sequence whose terms are 1 , 1 , 2 , 3 , 5 , 8 , 13 , . . . is called the Fibonacci sequence. This sequence has the property that, after starting with two 1’s, each term is the sum of the proceeding two. (a) Denoting the sequence by { a n } and starting with a 1 = a 2 = 1, show that a n +2 a n +1 = 1 + a n a n +1 if n 1. (b) Assuming that the sequence { a n +1 /a n } converges, find its limit L . As a cultural aside, it can be shown that a n = L n - (1 - L ) n 5 , a number one would hardly expect to even be an integer, let alone give us the Fibonacci sequence. (This is the part where your jaw falls off from the awesomeness of math.)
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